Published studies have shown that methane yield (g CH4/kg dry matter) from sheep is positively correlated with the size (volume and surface area) of the reticulo-rumen (RR) and the weight of its contents. However, the relationship between CH4 yield and RR shape has not been investigated. In this work, shape analysis has been performed on a data set of computerised tomography (CT) scans of the RR from sheep having high and low CH4 yields (n=20 and n=17, respectively). The three-dimensional geometries of the RRs were reconstructed from segmented scan data and split into three anatomical regions. An iterative fitting technique combining radial basis functions and principal component (PC) fitting was used to create a set of consistent landmarks which were then used as variables in a PC analysis to identify shape variation within the data. Significant size differences were detected for regions corresponding to the dorsal and ventral compartments between sheep with high and low CH4 yields. When the analysis was repeated after scaling the geometries to remove the effect of size, there was no significant shape variation correlating with CH4 yield. The results have demonstrated the feasibility of CT-based computational shape determination for studying the morphological characteristics of the RR and indicate that size, but not shape correlates with CH4 yield in sheep.

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case.

The understanding of the stock structure of a species is essential to effectively manage fisheries. Otolith shape analysis has been successfully applied in stock identification using the elliptic Fourier analysis. This method was used to assess possible differences in the otolith shape of Helicolenus dactylopterus caught around the Azores, Madeira and mainland Portugal. A total of 294 individuals ranging from 23 to 29 cm in total length, equally distributed by sex and area, were analysed. The multivariate analysis of variance performed on the otolith normalized elliptic Fourier descriptors (NEFDs) revealed significant differences for both area and sex, but no interaction was found between the two factors. In the canonical discriminant analysis an overall classification success of 69.4% and 66.7% was achieved for females and males respectively. These results suggest the usefulness of otolith shape analysis for stock differentiation of bluemouth from Portuguese waters.

Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

This review considers some recent advances in shape analysis based on landmark data, and focuses on the application of these methods to the study of skeletal evolution in primates. These advances have provoked some controversy. The major aims of this review are to put these debates in context and to provide an overview for the nonmathematician. The purpose of morphometric studies is considered, together with issues relating to the nature, significance and identification of landmarks before turning to a review of available technologies for the analysis of morphological variation. These are considered in terms of underlying models and assumptions in order to clarify when each is appropriate. To illustrate the application of these methods, 3 example studies are presented. The first examines differences amongst ancient and modern adult human crania using 2-dimensional data. The second illustrates the extension of these methods into 3 dimensions in a study of facial growth in monkeys. The third presents an application to the analysis of the form of the hominoid talus. The review ends with an account of available software resources for shape analysis.

Two techniques of stock discrimination both utilizing scales, i. e. termed as classical technique and shape analysis, are compared for discriminating between North American and European Atlantic salmon (Salmo salar L.) caught off West Greenland. Three scales from each of 60 salmon were analyzed: 30 from North American and 30 from European-origin salmon collected from home rivers in 1982. The best results indicated misclassification and error rates of 0% for shape analysis and 5.0% and ± 1.7%, respectively, for the classical technique. A nested ANOVA indicated that variation was greater between scales of salmon of North American and European origin than from several scales from the same salmon. It was concluded that shape analysis with computer-aided devices may be superior to the classical technique and it is recommended that it be examined for use at West Greenland.

Procedures are described for the acquisition and analysis of data in a study of the dental and facial characteristics of South Australian twins. Comparisons of the mesiodistal diameters of maxillary incisors in MZ and DZ twins revealed heterogeneity of total variances and evidence of inequality of mean values for some dimensions between MZ and DZ twins. Previous estimates of heritabilities for tooth size, relying on classical assumptions in twin research, may be exaggerated. A preliminary analysis of facial shape was undertaken using a procedure for shape matching based on a least squares fit of homologous coordinates. There was evidence of mirror-imaging in some MZ twin pairs and differences in facial asymmetry between male and female DZ twins. Future extensions of the study using methods for three-dimensional shape analysis are described.